Special Relativity Essentials

Concise answer

Two postulates — same physics in every inertial frame, same vacuum light speed for every observer — force moving clocks to run slow by $\gamma$ , moving lengths to contract by $\gamma$ , and energy and momentum to combine in the invariant $E^2 = (pc)^2 + (mc^2)^2$ .

Definitions

Inertial frame: A reference frame moving at constant velocity, in which Newton's first law holds; the postulates apply to all such frames.
Proper time: The time interval between two events measured in the frame where both events occur at the same location; it is the shortest measured interval.
Proper length: The length of an object measured in the frame where the object is at rest; it is the longest measured length.
Lorentz factor: The factor $\gamma = 1/\sqrt{1 - v^2/c^2}$, equal to 1 at rest and growing without bound as $v \to c$.
Rest energy: The energy $E_0 = mc^2$ an object has by virtue of its mass alone, in the frame where it is at rest.

Intuition

Both postulates sound harmless, but they collide: if everyone measures the same light speed, observers in relative motion cannot agree on both distances and times. Something has to give, and it is simultaneity — moving clocks tick slow and moving lengths shrink by exactly the factor that keeps $c$ the same for everyone.

The energy-momentum relation is a Pythagorean theorem: rest energy $mc^2$ and $pc$ are the legs, total energy $E$ is the hypotenuse. A massless particle has no rest-energy leg, so $E = pc$ ; a particle at rest has no momentum leg, so $E = mc^2$ .

Concept walkthrough

The first postulate says no inertial frame is special; the second says the vacuum speed of light is $c$ in every inertial frame, regardless of the motion of source or observer. Every kinematic result of special relativity follows from these two statements.

Time dilation: a clock at rest in one frame measures the proper time $\Delta\tau$ between two ticks; an observer watching that clock move at speed $v$ measures the longer interval $\Delta t = \gamma\Delta\tau$ . Length contraction is the flip side: the proper length $L_0$ is measured in the object's rest frame, and a moving observer measures the shorter $L = L_0/\gamma$ along the motion — transverse dimensions are untouched. The classic consistency check is the muon: its dilated lifetime (ground frame) and the contracted atmosphere (muon frame) predict the same survival.

Dynamics replaces $p = mv$ with $p = \gamma mv$ and defines total energy $E = \gamma mc^2$ , so kinetic energy is $K = (\gamma - 1)mc^2$ rather than $\tfrac{1}{2}mv^2$ . The invariant combination $E^2 = (pc)^2 + (mc^2)^2$ holds in every frame: it gives $E = pc$ for photons, reduces to the classical formulas when $v \ll c$ , and is usually the fastest route between energy and momentum on an exam.

After this page, you should be able to

State the two postulates and explain why they force observers to disagree about time and length measurements.
Identify proper time and proper length, and apply the Lorentz factor in the correct direction.
Compute relativistic momentum and total energy, and use $E^2 = (pc)^2 + (mc^2)^2$ to relate them.
Check answers against the classical limit $v \ll c$ and the massless limit $E = pc$.

Formulas and assumptions

Lorentz factor

\gamma = \dfrac{1}{\sqrt{1 - v^2/c^2}}

Variables

gamma: dimensionless Lorentz factor, at least 1
v: relative speed in meters per second
c: speed of light in meters per second

Assumptions

The relative motion is between inertial frames.
gamma approaches 1 for v much less than c and diverges as v approaches c.

Time dilation

\Delta t = \gamma\,\Delta\tau

Variables

Delta tau: proper time between events in seconds
Delta t: time interval measured by the observer who sees the clock move, in seconds

Assumptions

The proper time is measured in the frame where both events occur at the same place.
The moving clock is always measured to run slow, never fast.

Length contraction

L = \dfrac{L_0}{\gamma}

Variables

L_0: proper length in the object's rest frame in meters
L: length measured by an observer who sees the object move, in meters

Assumptions

Contraction applies only along the direction of relative motion.
Transverse dimensions are unchanged.

Relativistic momentum

p = \gamma m v

Variables

p: momentum in kilogram-meters per second
m: rest mass in kilograms
v: speed in meters per second

Assumptions

m is the invariant rest mass.
Reduces to p = mv when v is much less than c.

Energy-momentum relation

E^2 = (pc)^2 + (mc^2)^2

Variables

E: total energy in joules
p: momentum in kilogram-meters per second
m: rest mass in kilograms
c: speed of light

Assumptions

Holds in every inertial frame; the rest mass is invariant.
For massless particles it reduces to E = pc; total energy also equals gamma m c^2.

Worked example

Muon decay at 0.99c

A muon has a proper (rest-frame) mean lifetime of $2.2\,\mu\text{s}$ and travels at $v = 0.99c$ . Find its mean lifetime in the ground frame and the mean distance it travels before decaying.

1Lorentz factor: $1 - v^2/c^2 = 1 - (0.99)^2 = 1 - 0.9801 = 0.0199$ , so $\gamma = 1/\sqrt{0.0199} \approx 7.1$ .
2Dilated lifetime: $\Delta t = \gamma\Delta\tau \approx (7.1)(2.2\,\mu\text{s}) \approx 15.6\,\mu\text{s}$ .
3Distance in the ground frame: $d = v\Delta t = (0.99)(3.00\times 10^8\,\text{m/s})(15.6\times 10^{-6}\,\text{s}) \approx 4.6\times 10^3\,\text{m}$ , about 4.6 km.
4Contrast with the naive classical estimate $v\Delta\tau = (2.97\times 10^8)(2.2\times 10^{-6}) \approx 650\,\text{m}$ — time dilation is why muons created high in the atmosphere reach the ground.

The ground-frame lifetime is about $15.6\,\mu\text{s}$ and the muon travels roughly $4.6\,\text{km}$ on average, far beyond the classical 650 m estimate.

Common traps

Multiplying by $\gamma$ in the wrong direction; proper time is the shortest interval and proper length the longest, so check the answer's direction against that fact.
Applying length contraction to dimensions perpendicular to the motion; only the along-motion length contracts.
Using $K = \tfrac{1}{2}mv^2$ near light speed instead of $K = (\gamma - 1)mc^2$ .
Treating $E = mc^2$ as the total energy of a moving particle; it is the rest energy, and the total is $\gamma mc^2$ .
Forgetting the massless limit: photons obey $E = pc$ exactly, so they carry momentum despite having no mass.

GRE Physics overview Energy and Momentum Conservation Newton's Second Law and Free-Body Diagrams Quantum Mechanics Foundations Sign in for current practice

Question depth and domain coverage vary by exam. Practice answers are checked after submission.

Sources

GRE Subject Test Content and Structure — ETS. Accessed 2026-07-06. Use as a cited source for exam facts; do not imply affiliation or reproduce protected test material.
University Physics Volume 3, Section 5.1: Invariance of Physical Laws — OpenStax. Accessed 2026-08-02. OpenStax textbook content is CC BY-NC-SA 4.0; attribute and avoid verbatim reuse beyond short cited references.
University Physics Volume 3, Section 5.3: Time Dilation — OpenStax. Accessed 2026-08-02. OpenStax textbook content is CC BY-NC-SA 4.0; attribute and avoid verbatim reuse beyond short cited references.
University Physics Volume 3, Section 5.4: Length Contraction — OpenStax. Accessed 2026-08-02. OpenStax textbook content is CC BY-NC-SA 4.0; attribute and avoid verbatim reuse beyond short cited references.
University Physics Volume 3, Section 5.9: Relativistic Energy — OpenStax. Accessed 2026-08-02. OpenStax textbook content is CC BY-NC-SA 4.0; attribute and avoid verbatim reuse beyond short cited references.

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Last source check: 2026-08-02
Next scheduled review: 2026-11-02

Recheck ETS content areas and the OpenStax relativity references before each major GRE Physics preparation cycle.

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Special Relativity Essentials

Concise answer

Definitions

Intuition

Concept walkthrough

After this page, you should be able to

Formulas and assumptions

Lorentz factor

Time dilation

Length contraction

Relativistic momentum

Energy-momentum relation

Worked example

Muon decay at 0.99c

Common traps

Related pages and practice

Sources

Review and maintenance